measure theory – Feynman Path Integral and Wilsonian Renormalization

Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures.

The usual problem of Quantum Field Theory is to make sense of Feynman Path Integrals like

begin{equation}
int exp left(- frac{m^2}{2} int phi^2 + frac{Z}{2} int phi Delta phi – lambda int phi^4 right) mathrm{d} phi
end{equation}

where the outer integral is supposed to run over some space of test functions.
It is well-known that this does not really work.
What one can do, however, is for $lambda = 0$ to produce a “cylinder set probability measure” on e.g the space $mathcal{S}$ of Schwartz functions. More generally this works for any continuous positive definite bilinear form on $mathcal{S}$ and the resulting cylinder set measure may be pushed forward to $mathcal{S}’$ where it miraculously becomes a true Gaussian probability measure. Hence, for such theories we know what we are doing!

With $lambda neq 0$ this is no longer the case. In order to ensure the finiteness of expressions of the form $int phi^4$ we would like $phi$ to live in $mathcal{S}$.
To this end, we pick two non-negative mollifiers $chi, xi in mathcal{D}$ with $xi left( 0 right) = 1$. Furthermore, for any $epsilon > 0$ and $Lambda > 0$, define

begin{aligned}
chi_epsilon left( x right) &= epsilon^{-d} chi left( frac{x}{epsilon} right) \
xi_Lambda left( x right) &= xi left( frac{x}{Lambda} right) \
end{aligned}

and consider the continuous (? – it should certainly be Borel measurable) mapping

begin{aligned}
mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) : mathcal{S}’ &to mathcal{S} \
T &mapsto xi_Lambda cdot left( chi_epsilon ast T right) , .
end{aligned}

We may now take a Gaussian measure $mu$ on $mathcal{S}’$ corresponding to some bilinear theory and regularize it as $left( mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) right)_ast mu$ to a measure on $mathcal{S}$.
One can then define measures

begin{equation}
nu_{epsilon, Lambda} = exp left( -lambda int phi^4 right) cdot left( mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) right)_ast mu
end{equation}

normalize them, push them to $mathcal{S}’$ again and study their behaviour as $ left( epsilon, Lambda right) to left( 0, infty right)$.
But this also does not work as it just produces the expected divergences coming from attempting to multiply distributions.
Instead, one has to give up on keeping the model parameters $m, Z, lambda$ constant and let them depend on $epsilon$ and $Lambda$ (i.e on cutoffs) instead. One might then hope that for some nice $left( epsilon, Lambda right)$-dependence a limit (or at least a cluster point) $nu_{0, infty}$ indeed exists.

From a Wilsonian point of view, we would however like to split this $nu_{0, infty}$ into “fast” and “slow” modes depending on $epsilon$ and $Lambda$. That is, we want to fix a collection $left( mu_{epsilon, Lambda} right)_{epsilon ge 0, Lambda le infty}$ of probability measures on $mathcal{S}’$ such that for all $epsilon, delta > 0$ and all $Lambda, kappa > 0$

  • $nu_{0, infty}$ factorizes over $mu_{epsilon, Lambda}$, i.e $nu_{0, infty} = tilde{nu}_{epsilon, Lambda} ast mu_{epsilon, Lambda}$ for some probability measure $tilde{nu}_{epsilon, Lambda}$ on $mathcal{S}’$
  • $mu_{epsilon, Lambda} ast mu_{delta, kappa} = mu_{epsilon+delta, frac{lambda kappa}{Lambda + kappa}}$ (or someway similar)
  • $lim_{epsilon to 0, Lambda to infty} mu_{epsilon, Lambda} = mu_{0, infty}$

where somehow $nu_epsilon$ and $tilde{nu}_epsilon$ should be related.
Then we obtain a renormalization group equation

begin{equation}
tilde{nu}_{epsilon, Lambda} = tilde{nu}_{epsilon + delta, frac{lambda kappa}{Lambda + kappa}} ast mu_{delta, kappa}
end{equation}

as well as

begin{equation}
tilde{nu}_{0, infty} = mu_{0, infty} ast lim_{epsilon to 0, Lambda to infty} tilde{nu}_{epsilon, Lambda}
end{equation}

But, presuming the existence of some nice QFT corresponding to $nu_0$

  • Why would such a decomposition with respect to a such a family $mu_{epsilon, Lambda}$ exist? i.e why can we separate fast and slow modes in an arbitrary way?
  • How do we relate $nu_{epsilon, Lambda}$ and $tilde{nu}_{epsilon, Lambda}$ i.e how do we understand the map from bare to renormalized quantities and/or vice versa?

PS: The above is pretty much as far as my level of mathematics goes. Please try to not go too much further.